Trouble with Local Space Limit Rotation
Object constraints don't handle local space correctly when they don't have a parent.
But, that's a problem with an easy fix, because objects that are unparented exist in world space.  So for your unparented object, you can change the space in which the Limit Rotation acts from local space to world space.
You specifically ask how to limit in local space, although it's not clear what your end goal or actual problem is-- this looks like an idealized situation.  And it would be easy to imagine some misunderstanding of what limited local space rotation would look like.  If you want an explicit answer to how to limit rotation in local space, then make an empty at the world origin and parent your object to it.  Now, you can limit rotation in local space if you want.  It's just, that's exactly the same as limiting rotation in world space.